In generally you have two ip address one in internal ip address that is on your network and domain. another and another one is external ip address which is what you use to connect internet.
Now most internal addresses are either static or automatically assigned by dynamic. External addresses are usually assigned by the ISP, external addresses can be either Static or Dynamic depending on your service. Cable/Dedicated line providers are 99% static unless otherwise specified (it will never change short of you requesting it), while dialup/DSL are dynamic (unless otherwise specified).
On the internal network side of the router your IP address will only change if it is hard-wired with an Ethernet wire then you physically have to move the connector from 1 port to another as they are a fixed IP address (can be changed from within the router's set-up though).